(a) What is the impulse response
h
(
t
) of the ideal bandstop filter?
(b) Describe how to implement the ideal bandstop filter using only lowpass and highpass filters.
2.6.12 The ideal continuoustime bandpass filter has the frequency response
H
f
(
ω
) =
0

ω
 ≤
ω
1
1
ω
1
<

ω

< ω
2
0

ω
 ≥
ω
2
.
Describe how to implement the ideal bandpass using two ideal lowpass filters with different cutoff
frequencies. (Should a parallel or cascade structure be used?) Using that, find the impulse response
h
(
t
) of the ideal bandpass filter.
2.6.13 The ideal continuoustime highpass filter has the frequency response
H
f
(
ω
) =
0
,

ω
 ≤
ω
c
1
,

ω

> ω
c
.
Find the impulse response
h
(
t
).
2.6.14 The frequency response of a continuoustime LTI is given by
H
f
(
ω
) =
0

ω

<
2
π
1

ω
 ≥
2
π
(This is an ideal highpass filter.) Use the Fourier transform to find the output
y
(
t
) when the input
x
(
t
) is given by
(a)
x
(
t
) = sinc(
t
) =
sin(
π t
)
π t
(b)
x
(
t
) = sinc(3
t
) =
sin(3
π t
)
3
π t
2.6.15 What is the Fourier transform of
x
(
t
)?
x
(
t
) = cos(0
.
3
π t
)
·
sin(0
.
1
π t
)
X
f
(
ω
) =
F{
x
(
t
)
}
= ?
57
2.7
Matching
2.7.1 Match the impulse response
h
(
t
) of a continuoustime LTI system with the correct plot of its frequency
response

H
f
(
ω
)

.
Explain how you obtain your answer.
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
1
0.5
0
0.5
1
t
IMPULSE RESPONSE
8
6
4
2
0
2
4
6
8
0
0.2
0.4
0.6
0.8
1
ω
FREQUENCY RESPONSE A
8
6
4
2
0
2
4
6
8
0
0.2
0.4
0.6
0.8
1
ω
FREQUENCY RESPONSE B
8
6
4
2
0
2
4
6
8
0
0.2
0.4
0.6
0.8
1
ω
FREQUENCY RESPONSE C
8
6
4
2
0
2
4
6
8
0
0.2
0.4
0.6
0.8
1
ω
FREQUENCY RESPONSE D
58
2.7.2 For the impulse response
h
(
t
) illustrated in the previous problem, identify the correct diagram of the
poles of
H
(
s
).
Explain how you obtain your answer.
3
2
1
0
1
2
3
10
5
0
5
10
#1
Real part
Imag part
3
2
1
0
1
2
3
10
5
0
5
10
#2
Real part
Imag part
3
2
1
0
1
2
3
10
5
0
5
10
#3
Real part
Imag part
3
2
1
0
1
2
3
10
5
0
5
10
#4
Real part
Imag part
59
2.7.3 The following diagrams indicate the pole locations of six continuoustime LTI systems.
Match each
with the corresponding impulse response with out actually computing the Laplace transform.
POLEZERO DIAGRAM
IMPULSE RESPONSE
1
2
3
4
5
6
1
0
1
6
4
2
0
2
4
6
#1
1
0
1
6
4
2
0
2
4
6
#2
1
0
1
6
4
2
0
2
4
6
#3
1
0
1
6
4
2
0
2
4
6
#4
1
0
1
6
4
2
0
2
4
6
#5
1
0
1
6
4
2
0
2
4
6
#6
0
2
4
1
0
1
2
A
0
2
4
1
0
1
2
B
0
2
4
1
0
1
2
C
0
2
4
1
0
1
2
D
0
2
4
1
0
1
2
E
0
2
4
1
0
1
2
F
60
2.7.4 The figure shows the impulse responses and frequency responses of four continuoustime LTI systems.
But they are out of order.
Match the impulse response to its frequency response magnitude, and
explain
your answer.
IMPULSE RESPONSE
FREQUENCY RESPONSE
1
2
3
4
1
0
1
2
3
4
5
6
1
0.5
0
0.5
1
IMPULSE RESPONSE 1
10
5
0
5
10
0
0.2
0.4
0.6
FREQUENCY RESPONSE 1
1
0
1
2
3
4
5
6
1
0.5
0
0.5
1
IMPULSE RESPONSE 2
10
5
0
5
10
0
0.2
0.4
0.6
FREQUENCY RESPONSE 3
1
0
1
2
3
4
5
6
1
0.5
0
0.5
1
IMPULSE RESPONSE 3
10
5
0
5
10
0
0.2
0.4
0.6
FREQUENCY RESPONSE 2
1
0
1
2
3
4
5
6
1
0.5
0
0.5
1
IMPULSE RESPONSE 4
TIME (SEC)
10
5
0
5
10
0
0.2
0.4
0.6
FREQUENCY RESPONSE 4
FREQUENCY (Hz)
61
2.7.5 The figure shows the impulse responses and frequency responses of four continuoustime LTI systems.
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 Summer '19
 Signal Processing, LTI system theory, Impulse response, CASCADE